Anthony W. Knapp

Lie Groups Beyond an Introduction

Birkhauser Boston • Basel • Berlin CONTENTS

List of Figures x Preface xi Prerequisites by Chapter xiv Standard Notation xv

I. LIE ALGEBRAS AND LIE GROUPS 1 1. Definitions and Examples 2 2. Ideals 7 3. Field Extensions and the Killing Form 11 4. Semidirect Products of Lie Algebras 15 5. Solvable Lie Algebras and Lie's Theorem 17 6. Nilpotent Lie Algebras and Engel's Theorem 22 7. Cartan's Criterion for Semisimplicity 24 8. Examples of Semisimple Lie Algebras 31 9. Representations of s 1(2, C) 37 10. Elementary Theory of Lie Groups 43 11. Automorphisms and Derivations 55 12. Semidirect Products of Lie Groups 58 13. Nilpotent Lie Groups 62 14. Classical Semisimple Lie Groups 66 15. Problems 73

II. COMPLEX SEMISIMPLE LIE ALGEBRAS 79 1. Classical Root Space Decompositions 80 2. Existence of Cartan Subalgebras 85 3. Uniqueness of Cartan Subalgebras 92 4. Roots 94 5. Abstract Root Systems 103 6. Weyl Group 116 7. Classification of Abstract Cartan Matrices 123 8. Classification of Nonreduced Abstract Root Systems 138 9. Serre Relations 139 10. Isomorphism Theorem 149 11. Existence Theorem 152 12. Problems 156

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III. UNIVERSAL ENVELOPING ALGEBRA 164 1. Universal Mapping Property 164 2. Poincare-Birkhoff-Witt Theorem 168 3. Associated Graded Algebra 172 4. Free Lie Algebras 177 5. Problems 179

IV. COMPACT LIE GROUPS 181 1. Examples of Representations 181 2. Abstract Representation Theory 186 3. Peter-Weyl Theorem 191 4. Compact Lie Algebras 196 5. CentralizersofTori 198 6. Analytic Weyl Group 206 7. Integral Forms 210 8. Weyl's Theorem 214 9. Problems 215

V. FINITE-DIMENSIONAL REPRESENTATIONS 219 1. Weights 220 2. Theorem of the Highest Weight 225 3. Verma Modules 229 4. Complete Reducibility 236 5. Harish-Chandra Isomorphism 246 6. Weyl Character Formula 259 7. Parabolic Subalgebras 269 8. Application to Compact Lie Groups 277 9. Problems 283

VI. STRUCTURE THEORY OF SEMISIMPLE GROUPS 291 1. Existence of a Compact Real Form 292 2. Cartan Decomposition on the Lie Algebra Level 298 3. Cartan Decomposition on the Lie Group Level 304 4. Iwasawa Decomposition 311 5. Uniqueness Properties of the Iwasawa Decomposition 320 6. Cartan Subalgebras 326 7. Cayley Transforms 330 8. Vogan Diagrams 339 9. Complexification of a Simple Real Lie Algebra 348 10. Classification of Simple Real Lie Algebras 349 11. Restricted Roots in the Classification 362 12. Problems 367 Contents ix

VII. ADVANCED STRUCTURE THEORY 372 1. Further Properties of Compact Real Forms 373 2. Reductive Lie Groups 384 3. KAK Decomposition 396 4. Bruhat Decomposition 397 5. Structure of M 401 6. Real-Rank-One Subgroups 408 7. Parabolic Subgroups 411 8. Cartan Subgroups 424 9. Harish-Chandra Decomposition 435 10. Problems 450

VIII. INTEGRATION 456 1. Differential Forms and Measure Zero 456 2. Haar Measure for Lie Groups 463 3. Decompositions of Haar Measure 468 4. Application to Reductive Lie Groups 472 5. Weyl Integration Formula 479 6. Problems 485

APPENDICES A. Tensors, Filtrations, and Gradings 1. Tensor Algebra 487 2. Symmetric Algebra 492 3. Exterior Algebra 498 4. Filtrations and Gradings 501 B. Lie's Third Theorem 1. Levi Decomposition 504 2. Lie's Third Theorem 507 C. Data for Simple Lie Algebras 1. Classical Irreducible Reduced Root Systems 508 2. Exceptional Irreducible Reduced Root Systems 511 3. Classical Noncompact Simple Real Lie Algebras 518 4. Exceptional Noncompact Simple Real Lie Algebras 531

Hints for Solutions of Problems 545 Notes 565 References 585 Index of Notation 595 Index 599